Speech emotion recognition (SER) often experiences reduced performance due to background noise. In addition, making a prediction on signals with only background noise could undermine user trust in the system. In this study, we propose a Noise Robust Speech Emotion Recognition system, NRSER. NRSER employs speech enhancement (SE) to effectively reduce the noise in input signals. Then, the signal-to-noise-ratio (SNR)-level detection structure and waveform reconstitution strategy are introduced to reduce the negative impact of SE on speech signals with no or little background noise. Our experimental results show that NRSER can effectively improve the noise robustness of the SER system, including preventing the system from making emotion recognition on signals consisting solely of background noise. Moreover, the proposed SNR-level detection structure can be used individually for tasks such as data selection.